Reflection, refraction, and diffraction of optical wavesby fluid–fluid boundaries have received considerableattention recently in the studies of shocks in fluidsand of light scattering by fluid interfaces. For ex-ample, in 1995 Adamovsky and Johnson ~Universityof Akron! used a single scanning technique to study
ow visualization,1 and Panda ~University of Toledo!and Adamovsky investigated scattering by shocks.2In 1996 Rashidnia used vertical temperature gradi-ent methods in determining characteristics of ther-mocapillary migration speeds in water3 that followedother research in the area by Subramanian4 ~Clark-on University!.We applied single-beam and temperature gradientethods in studying scattered light from boundaries
n heated water. We now present additional resultsf the measurements reported in 1998 of intensitieshat are due to scattered pencil laser beams.5 Our
results are for beams grazing a bubble ~less than onethird of a beam striking the interface! and beams
S. Giles, Jr. [email protected]! is with the Departmentof Electrical Engineering and Computer Science, University ofToledo, Toledo, Ohio 43606. G. Adamovsky is with the Instru-mentation and Control Group, NASA Glenn Research Center,21000 Brookpark Road, Mail Stop 77-1, Cleveland, Ohio 44135.N. Rashidnia is with the National Center for Microgravity Re-search on Fluids and Combustion, NASA Glenn Research Centerat Lewis Field, 21000 Brookpark Road, Mail Stop 110-3, Cleveland,Ohio 44135.
Received 18 August 2000; revised manuscript received 12 March2001.
3190 APPLIED OPTICS y Vol. 40, No. 19 y 1 July 2001
glancing a bubble ~one third or more of the beamstriking the interface!. The angle of incidence waspproximately 90 deg for both grazing and glancingeams. We also present results for beams passinghrough the surrounding test fluid without strikinghe interface. It is presumed that the index of re-raction and temperature are related, as reported by
assoli et al. ~Consiglio Nazionale delle Ricerche!6 in1993, and that the water is slightly compressible, asindicated in the density data of Scott and Bigg7 pre-
ared for the National Research Council in 1928.Langley ~St. Johns University! and Marston
~Washington State University! investigated lightscattering by spherical air bubbles in water underisothermal conditions.8 In the present study, weconsider the effects of forward-scattered light in dis-tilled water having various temperature gradientsincluding the near-isothermal condition. We de-tected changes in the refractive index when the tem-perature changed and when the temperaturegradient changed. Indirectly, we detected effects ofsmall changes in refractive indices using photo-graphic data obtained from forward signals.
2. Background Theory
We assume that a light beam travels in the positive zdirection in air and normally strikes a rectangularparallelepiped glass container of water. At approx-imately a normal angle of incidence, the beam strikesthe water–glass planar boundary ~located at z 5 0!.t then travels through the water. In this simpleodel we assume that the water is homogeneous in
he z and y directions, but is nonhomogeneous in thedirection because of a temperature gradient from its
Fd
wttudCBt
6
pts3wsfiWtesaps
top ~x 5 0! to the bottom ~x 5 260 mm!. No airbubble is present in the water in this simple model.
In such a lossless, nonmagnetic, source-free, linear,isotropic, and stationary medium, the index of refrac-tion m is a continuous function of one variable: theheight position x. We assumed planar layers each ofinfinitesimal thickness in the x direction. Accordingto Pollock,9 the eikonal ray path for this model obeysthe following equation:
d2xydz2 5 m21~ x!dmydx. (1)
or distilled water, m changes with fluctuations inensity caused by temperature gradients.10,11 From
Eisenberg’s research,11 the index of refraction de-pends on temperature and density through the f func-tion of the following equation:
m2 5 ~1 1 2f !y~1 2 f !, (2)
here f 5 ArB exp~2CT!. Here T is the Celsiusemperature; r is the density in grams per millili-er; and A, B, and C are constants of appropriatenits. It is assumed that the temperature andensity are straight-line functions of x. ~For thelausius–Mossotti model of the fluid, the constantsand C are 1 and 0, respectively.! Interpolating
he results of Table IIa of Eisenberg11 for our case atl 5 632.8 nm, we obtain the numeric values of theconstants as A 5 0.2057945, B 5 0.88639, and C 5
.0808 3 1025.A particular solution within an additive constant of
the above ordinary second-order differential equationis
* dxy~2 ln m!1y2 5 z. (3)
We integrated Eq. ~3! using MATLAB for three temper-atures at the top of the test chamber: 60 °C, 40 °C,and 20 °C. For all top temperatures, the quadratureintegration produced the same results to within thefourth place to the right of the decimal point. Thetheoretical optical ray path bends toward cooler me-dia as indicated in Fig. 1 for the simple model.
A positive average temperature gradient resultedfrom a higher temperature at the top and a lowerwater temperature at the bottom. For positive tem-perature gradients, we expected the bending to bevertically downward, with the bottom temperaturefixed at 20 °C. Our experiments confirmed this ba-sic result. Moreover, we used these deflections todetect temperature gradients or, equivalently, smallchanges in indices of refraction. We used resultsfrom two other cases: the less-known case of up-ward bending of beams striking a bubble in the morevolatile, less predictive environment to detect nega-tive gradients; and the well-known case of reflectionsand refraction by air bubbles in water to detectabrupt boundaries.
3. Optical Setup
Figure 2 shows the experimental setup. In addition
to two He–Ne laser sources, the central item for oursystem was a bubble, drop, and particle unit device, achamber that contained the test fluid. The test cell~chamber! was 45 mm 3 45 mm 3 60 mm. Water,the fluid under test, occupied this volume enclosed byBK-7 glass on the side faces and anodized aluminumon the top and bottom faces. Using a syringe, weintroduced air through a small hole in the center ofthe top of the test cell forming a bubble on the innersurface of the anodized aluminum.
We used collimated light ~from laser 1, a type IIIblaser! to view the abrupt interface and the relative
osition of the striking laser beam ~from laser 2 ofype IIIa!. The collimator consisted of a 403 micro-cope objective and an infinity-focused double-convex-in. ~8-cm! lens. The 0.8-mm-wide beam of laser 2as raised until approximately 50% of the beam
truck the bubble interface producing reflected, re-racted, and diffracted light. In the gradual diffusednterface case, the entire beam bypassed the bubble.
e recorded the resulting scattered light from theest chamber. Pictures were taken with a CCD cam-ra after light passed through a semitransparentcreen ~Engineering Pad paper 17-450-08!. The im-ge taken with the mobile cooled CCD camera wasrocessed with PMIS image processing computeroftware. In addition, for each experimental run, a
Fig. 1. Plot of ideal ray paths for three temperature gradients intest chamber H2O ~20y20, 40y40, and 60y20!.
Fig. 2. Optical setup.
1 July 2001 y Vol. 40, No. 19 y APPLIED OPTICS 3191
aFrwgescf
v6ip
3
photograph of scattered light was taken with an or-dinary film camera.
We maintained the average temperature gradientby using two thermostatic circulators. One circula-tor maintained a constant temperature at the topaluminum surface and the other at the bottom sur-face. The top and bottom sections of the test cham-ber were equipped with thermocouples adjacent tothe cell volume. Circulating coolant occupied adja-cent volumes above and below the test cell with tem-peratures measured with an Omega thermocoupledthermometer.
Measurements were obtained from a leveled facil-ity. The parallel laser beams defined a plane per-pendicular to the gravity field. Both the positions ofthe laser 2 Gaussian beam and of the CCD camerawere measured with respect to the position of thelaser 1 beam between the near-CCD camera positionand the beam-splitter position when no test chamberwas present. The test chamber was placed in thefacility after beam calibrations were performed.
4. Measured Results
Laser 2 beam intensities were measured with a pow-ermeter at three positions: ~A! 7.5 cm before the testchamber, ~B! 17.5 cm after the test chamber, and ~C!47.5 cm after the test chamber and before the CCDcamera. Figure 3 shows these power density mea-surements for the temperatures considered and thebubble effects on light intensity.
An approximately hemispherical air–water bubblewas introduced. At the location immediately after
Fig. 3. Plot of beam intensities at three locations for the cases ofat 20 °C.
192 APPLIED OPTICS y Vol. 40, No. 19 y 1 July 2001
the test chamber, the largest cross-sectional area ofthe bubble ~with respect to the direction of the beam!was comparable with the expanded cross-sectionalarea of the beam. We measured the height andwidth of the bubble with the laser beam utilizing theshadowgraph; we measured its volume by weighingthe displaced water caused by the bubble’s injection.Typical height and width measurements were 4.386and 11.085 mm, respectively. Typical volume andexpanded laser 2 beam diameter measurements were0.393 cm3 and 3.018 mm .. l, respectively.
The laser 2 beam was positioned to strike the poleof the bubble. Figure 4 shows ray paths caused byreflections and refractions by an almost hemispheri-cal bubble. The resulting interference fringes in theforward direction were recorded along with all otherscattered light. An extended version of the darkcross pattern ~noted by Langley and Marston8! waspparent from the photographs. As can be seen inig. 5, there is an elongated sector in the direction ofay bending. ~The bright spots near the main beamere due to higher-order reflections from the BK-7lass. Dark fringes through these reflections werenhanced.! We called this elongated sector of light atreak. A similar streak is apparent when the CCDamera is in the far-field location ~1.4 m farther awayrom the test chamber!.
Typical graphs of intensities across the beams forarious temperature gradients are shown in Figs.–10. They show the intensity along a vertical line
n the streaks at the column location of the brightestixel. ~The row pixel numbers correspond to vertical
striking and not striking the bubble. The chamber bottom was
light
measurements; downward is positive. Approxi-mately 10.9 image pixels are equivalent to 1 mm ofactual displacement on the CCD region of interest.!Figures 6–10 show results for the top and bottomtemperatures in the near field and far field for beamsstriking the bubble and not striking the bubble. Ascan be seen in the graphs, the light intensity variedwith position over the relevant cross-sectional areainside the scattered beam.
5. Other Results
A. Abrupt Interfaces
The abrupt interface can be viewed as an anomalousdiffraction problem. Both the size factor d 5 phmyland the phase factor p 5 2d~m 2 1! are much greaterthan unity, where h is the height of the bubble andthe index of refraction m ' 1.33. Thus both thefirst- and the second-order scattered signals are sig-nificant.12 Geometrical optics largely explains the
Fig. 4. Reflected and refracted ray paths produced by an incidentray.
Fig. 5. Forward-scattered light from a beam striking a bubble.
Fig. 6. CCD-measured beam intensity at a 20y20 gradient strik-ing a bubble in the ~a! near field and ~b! far field.
Fig. 7. CCD-measured beam intensity at a 20y20 gradient notstriking a bubble in the ~a! near field and ~b! far field.
1 July 2001 y Vol. 40, No. 19 y APPLIED OPTICS 3193
i
3194 APPLIED OPTICS y Vol. 40, No. 19 y 1 July 2001
bright streaks. These streaks are apparent in theCCD camera images well below the beam positionbecause there is a nonnoise signal on each pixel rowbelow the beam. That is, where the intensity versescolumn pixel is plotted, there is a nonnoise signalassociated with each row below the beam down to row255. In addition, the row signals far above the beamare nonintelligent signals. However, above theglancing beam but near it there is an intelligent sig-nal best described by physical optics.
Photographs of forward-scattered signals afterstriking the bubble indicate the two major effects.The geometrical-optics streak is indicated by the raysof Fig. 4. The appearance of the second-order effectsof anomalous diffraction is shown in Fig. 5; intelligentsignals in the region above the beam—but near it—are apparent.
The streak began when the rays of the beam grazedthe bubble and became more pronounced as the cen-ter of the beam glanced the interface. When thechange of refractive index was appreciable, thestreak’s intensity was greatest indicating the locationof the abrupt interface. The angle made by thestreak directly implied the angle that the interfacemade with the polar axis of the bubble.
B. Diffused Interfaces
An applied temperature gradient produced a nonho-mogeneous medium, even when a bubble was notpresent. The abrupt boundary ~the air–water inter-
Fig. 10. CCD-measured beam intensity at a 16y30 gradient strik-ng a bubble in the ~a! near field and ~b! far field.
Fig. 8. CCD-measured beam intensity at a 60y20 gradient strik-ing a bubble in the ~a! near field and ~b! far field.
Fig. 9. CCD-measured beam intensity at a 60y20 gradient notstriking a bubble in the ~a! near field and ~b! far field.
diTiabwttamac
rnTgtdicw
face, or bubble! produced a rapidly changing index ofrefraction near the boundary. The temperature gra-dient produced a continually changing index of re-fraction profile, or what we referred to as diffusedinterfaces.
First, we investigated light from the rapidly chang-ing diffused interfaces produced by the presence of abubble in a temperature gradient. Inspections ofthe fringe pattern near the abrupt interface revealedthe spacing of minima and maxima of an associatedintensity expansion function ~see Figs. 6–10!. The
istribution of the power in the bright rings below thenterface changed with temperature gradient.hese relative hot spots shifted lower as the gradient
ncreased ~with a positive gradient from top to bottomcross the test cell!. When the beam struck the bub-le in positive gradients, the distribution of intensityas downward with associated sidelobes at the bot-
om edge of the main pulse. The sidelobe ~betweenwo dark fringes! increased with the magnitude of theverage temperature gradient. Accurate measure-ents of shifts for large negative gradients were not
ttempted with bubbles present because of the diffi-ulties in maintaining the bubbles.
Second, information on the gradual change in theefractive index was contained in the fringe patternsear the hottest spot of the fully bypassing beam.he above-mentioned downward shift for positiveradients was confirmed. Moreover, it was shownhat the shift is upward for a negative gradient. Theistribution of intensity of the transmitted beam var-ed across the beam, and the rate of variationhanged with the temperature gradient shifting up-ard for the negative gradient.
6. Conclusions
There is current interest in the study of gaseous flowsin combustion processes in rocket and jet engines.Laser techniques allow the remote observation, asopposed to in situ observations, of these processes.We discussed how to detect changes in indices ofrefraction using a single laser beam instead of severalbeams. A beam was used to produce scattered lightfrom boundaries in water caused by temperature gra-dients and an attached air bubble. The intensity ofthe forward-collected light was used to determinechanges in the refractive indices. Reflected and re-fracted light was used to detect abrupt interfaces.
We have presented new results using a single-beam experimental technique. The experiment con-firmed the bending of a light beam in water towardcooler regions as predicted by use of the empiricalresults of Eisenberg. It also showed that a forward-scattered beam has an intensity profile that varieswith the applied temperature gradient. We ob-served both first- and second-order effects using CCDphotographs of laser signals striking an air–waterbubble.
This research was supported by a National Aero-nautics and Space Administration Summer FacultyFellowship program at the Glenn Research Center.The authors thank each of the reviewers of the paper.
References1. G. Adamovsky and D. K. Johnson, “Optical techniques for
shock visualization and detection,” in Optical Techniques inFluid, Thermal, and Combustion Flow, S. S. Cha and J. D.Trolinger, eds., SPIE 2546, 348–357 ~1995!.
2. J. Panda and G. Adamovsky, “Laser light scattering by shockwaves,” Phys. Fluids 7, 2271–2279 ~1995!.
3. N. Rashidnia, “Instabilities around a bubble due to combinedMarangoni and buoyancy effects,” AIChE Symp. Ser. 92, 110–118 ~1996!.
4. R. S. Subramanian, “The motion of bubbles and drops in re-duced gravity,” in Transport Processes in Drops, Bubbles, andParticles, R. Chhabra and D. Dekee, eds. ~Hemisphere, NewYork, 1992!, pp. 1–42.
5. G. Adamovsky and S. Giles, “Laser pencil beam based tech-niques for visualization and analysis of interfaces betweenmedia,” NASA Tech. Memo 206635 ~Glenn Research Center,Cleveland, Ohio, 1998!, pp. 1–6.
6. P. Massoli, F. Beretta, A. D’Alessio, and M. Lazzaro, “Temper-ature and size of single transparent droplets by light scatteringin the forward and rainbow regions,” Appl. Opt. 32, 3295–3301~1993!.
7. V. Scott and P. H. Bigg, “Density and specific volume of water,”in International Critical Tables of Numerical Data, Physics,Chemistry and Technology, C. J. West and N. E. Dorsey, eds.~McGraw-Hill, New York, 1928!, Vol. 3, pp. 24–26.
8. D. S. Langley and P. L. Marston, “Forward glory scatteringfrom bubbles,” Appl. Opt. 30, 3452–3458 ~1991!.
9. C. R. Pollock, Fundamentals of Optoelectronics ~Irwin, Chi-cago, Ill., 1995!, p. 108.
10. M. Kerker, The Scattering of Light and Other ElectromagneticRadiation ~Academic, New York, 1969!, pp. 487–498.
11. H. Eisenberg, “Equation of the refractive index of water,”J. Chem. Phys. 43, 3887–3892 ~1965!.
12. H. C. van de Hulst, Light Scattering by Small Particles ~Dover,New York, 1981!, p. 173.
1 July 2001 y Vol. 40, No. 19 y APPLIED OPTICS 3195
